← summary

What sets the homogenization scale?

Ergodicity gives averaging. The covariance tail decides the size of the first fluctuation correction.

A fast finite-state chain has an integrable covariance, so its symmetric correction is proportional to its correlation time. That conclusion is not a consequence of stationarity and ergodicity alone.

Long memory changes the power of the small parameter. Conversely, a vanishing Green–Kubo integral can make averaging faster than the usual diffusive rate, while a periodic ergodic driver can have no convergent Green–Kubo integral at all.

The exact starting point

Let $X_t$ be a centered stationary scalar process with covariance $C(t)=\mathbb E[X_0X_t]$, and accelerate it to $X_{t/\varepsilon}$. For

stationarity alone gives the exact identity

Everything below follows from this formula or its equivalent spectral form. Ergodicity is relevant to the averaged limit, but it does not determine the rate at which the variance vanishes.

The low-frequency theorem

The covariance-tail results below require eventual sign and regular variation in the time domain. A spectral formulation removes that restriction. Write Bochner's one-sided representation as

Fubini's theorem gives the exact Fejér-kernel identity

The atom is not optional under stationarity alone. The continuous-time mean ergodic theorem identifies

where $P_{\rm inv}$ is projection onto the shift-invariant subspace. Indeed, with $R=T/\varepsilon$,

Thus a centered ergodic process has $\mu(\{0\})=0$. More generally, zero mass at frequency zero is exactly the condition needed for this particular observable to average in mean square; it does not by itself imply that the whole stationary system is ergodic.

If $\mu(\{0\})>0$, the variance converges to $T^2\mu(\{0\})$ and no fluctuation rescaling can repair the failed averaging limit. This is the spectral projection underlying von Neumann's mean ergodic theorem (1932).

The full maturity/switching crossover

The exact identity also classifies every relative maturity scale, without regular variation. Define

Then

Because $0\leq K_R\leq1$, dominated convergence gives the complete trichotomy:

The proof is only pointwise kernel convergence: $K_R\to1$ as $R\to0$, $K_R\to K_r$ as $R\to r$, and $K_R\to\mathbf1_{\{0\}}$ as $R\to\infty$. Thus $T_\varepsilon\to0$ by itself does not imply a frozen or homogenized limit: the decisive quantity is the number $T_\varepsilon/\varepsilon$ of fast correlation times observed. The regularly varying theorem below is the quantitative refinement of the third row when $\mu(\{0\})=0$.

Vector covariance and the rank caveat

The same crossover holds for a centered second-order stationary vector $X_t\in\mathbb R^d$. Let $\mathsf F$ be its finite nonnegative Hermitian matrix-valued spectral measure on $\mathbb R$, the matrix extension of Cramér's spectral representation (1940); Wiener and Masani (1958) give the classical multivariate theory. For the vector integral $A_\varepsilon(T)$,

Applying scalar dominated convergence to $u^*\Sigma_Ru$ for every $u$ and then polarizing gives matrix-norm convergence

This does not make rank continuous or monotone in maturity. Let $U,V$ be independent uniform phases and

The irrational torus flow is stationary and ergodic, with covariance $\operatorname{diag}\{\cos t,\cos(\sqrt2t)\}$, hence

Its ranks at $R=\pi,2\pi,3\pi$ are respectively $2,1,2$: at $2\pi$ the first frequency lands exactly on a zero of the Fejér kernel while the irrational frequency does not. Moreover, $\Sigma_R\to0$ as $R\to\infty$ although its rank is two for generic finite $R$. This maturity-dependent spectral rank is distinct from the fixed-feature Green–Kubo rank bound for a finite-state driver; neither statement implies the other.

Assume now that $\mu(\{0\})=0$, put $F(x)=\mu((0,x])$, and suppose that the cumulative spectral mass is regularly varying:

where $L$ is slowly varying. No density is required. More generally, as $\varepsilon\downarrow0$, let the maturity $T_\varepsilon>0$ vary and assume only that

Then

This includes fixed maturity, shrinking maturity with many fast correlation times still observed, and growing maturity. For a general stationary process it is a variance theorem, not a functional limit theorem. For a stationary Gaussian driver, however, the variance result determines the entire limit law.

Matrix regular variation and a spectral rank hierarchy

The low-frequency theorem also has a finite-dimensional matrix form. For the vector spectral measure above, assume there is no atom at zero and define the symmetrized cumulative matrix

If, in operator norm,

then along every $R_\varepsilon=T_\varepsilon/\varepsilon\to\infty$,

in operator norm. This follows directly from the matrix Fejér identity: matrix Potter bounds control the small-frequency part uniformly, frequencies bounded away from zero are $o(T_\varepsilon^{2-\alpha}\varepsilon^\alpha L(R_\varepsilon))$, and finite-dimensional polarization identifies the limit.

Equivalently one may first test every quadratic form $u^*\mathsf M u$ and then use compactness of the unit sphere. The conclusion includes singular $B$; no positive-definiteness assumption is hidden in the proof.

That last point creates a genuine hierarchy. Let the two coordinates be independent stationary Gaussian processes with

Their one-sided spectral densities are $e^{-\omega}\omega^{\alpha_j-1}/\Gamma(\alpha_j)$. For every finite $\varepsilon>0$ the integrated covariance is diagonal and full rank, but its eigenvalues satisfy

Thus normalization at the dominant $\varepsilon^{0.6}$ scale produces a rank-one limit; the second direction is visible only at order $\varepsilon^{1.6}$, and the condition number grows like a known constant times $\varepsilon^{-1}$. This asymptotic spectral rank is not the exact finite-$\varepsilon$ rank and is not the fixed-feature Green–Kubo rank of a finite-state chain.

The exact Gamma-spectrum certificate measures eigenvalue orders $0.599999990$ and $1.599757660$. The two leading-constant ratios are $0.999999993$ and $0.999474581$, while the condition-number ratio to its predicted equivalent is $1.000525688$.

At the smallest tested $\varepsilon$, the eigenvalues normalized by $\varepsilon^{0.6}$ are $2.09923303$ and $2.5102\times10^{-8}$. Independent covariance quadrature agrees with the closed spectral formula within $4.45\times10^{-16}$.

Gaussian functional limit in the sharp subcritical Hölder topology

Let $X$ now be centered, stationary and Gaussian, and put

Under the same cumulative spectral hypothesis, set $H=1-\alpha/2\in(0,1)$.

For $0<\gamma<1$, let $c_0^\gamma[0,1]$ denote the separable little-Hölder space: paths $f(0)=0$ with finite $\gamma$-Hölder norm and

Then, as $R\to\infty$,

where $B_H$ is standard fractional Brownian motion. Equivalently, along every joint path $T_\varepsilon/\varepsilon\to\infty$, the process $A_\varepsilon(tT_\varepsilon)/\sqrt{\operatorname{Var}A_\varepsilon(T_\varepsilon)}$ has the same limit.

Indeed, the spectral theorem says that $V$ is regularly varying with index $2-\alpha=2H$. Stationary increments give the exact identity

which converges to $\{s^{2H}+t^{2H}-|t-s|^{2H}\}/2$. Gaussianity turns covariance convergence into finite-dimensional convergence.

The usual invocation of Potter bounds needs one extra step here: regular variation controls $V(Rh)/V(R)$ only when $Rh$ is large. Fix $\eta>0$. For $Rh$ above a fixed threshold, Potter's bound gives $V(Rh)/V(R)\le C_\eta h^{2H-\eta}$.

For microscopic lags, stationarity and $|\operatorname{Cov}(X_0,X_u)|\le\operatorname{Var}(X_0)$ give $V(r)\le\operatorname{Var}(X_0)r^2$. Since $2-2H+\eta>0$ and $R^{-\eta}/L(R)$ is bounded for slowly varying $L$, the same inequality follows uniformly for $Rh$ below the threshold. Thus

uniformly for all sufficiently large $R$. Gaussian moment equivalence and the Hölder version of Kolmogorov's criterion imply tightness in $c_0^\gamma$ after choosing $p$ and $\eta$ so that $\gamma<H-\eta/2-1/p$. Together with the covariance limit, this proves the displayed convergence.

The topology is sharp. Fractional Brownian motion has exact uniform modulus $h^H\sqrt{\log(1/h)}$ up to constants, so its paths are almost surely in every $c_0^\gamma$, $\gamma<H$, but not in $C^H$.

If the laws of $S_R$ were tight in $c_0^\gamma$ for any $\gamma\ge H$, continuous embedding into $C[0,1]$ and uniqueness of the finite-dimensional limit would produce an fBm-valued limit in that space, a contradiction. The modulus statement is also the ordinary-fBm specialization of Wang and Xiao's exact theorem (2022).

No mixing assumption is needed beyond the spectral regular variation, but Gaussianity is essential: the same variance asymptotic does not determine the limit for a general process. The long-memory half, $0<\alpha<1$, is the linear Gaussian case of Taqqu's weak-convergence theorem (1975); the limit is the fractional Brownian motion introduced by Mandelbrot and Van Ness (1968).

The exact Gamma-spectrum certificate checks the uniform variance envelope on a dyadic mesh extending ten scales below the correlation time. For $(\alpha,H,\gamma)=(0.7,0.65,0.55)$ its largest nontrivial ratio $\{V(Rh)/V(R)\}/h^{2\gamma}$ is $0.8705506$ through $R=2^{80}$, while the correlation-time ratio falls to $0.0000061$.

For $(1.7,0.15,0.10)$ the corresponding values are $0.9330330$ and $0.0003460$. The fitted correlation-time decay exponents are $0.2000000$ and $0.1000015$, against the predicted $2(H-\gamma)$ values $0.2$ and $0.1$.

The critical Gaussian boundary is not fractional Brownian motion

The restriction $\alpha<2$ in the functional theorem is sharp. At the divergent critical endpoint, suppose

The endpoint spectral theorem below gives $V(R)\sim4bH(R)$. The divergent integral $H$ is slowly varying, so $V(cR)/V(R)\to1$ for every fixed $c>0$. Hence for a centered stationary Gaussian driver the same exact stationary-increment identity used above yields, for $s,t>0$,

Thus on any finite collection of distinct positive times the limit has the representation $G(t_i)=(Z_0+Z_i)/\sqrt2$, where $Z_0,Z_1,\ldots$ are independent standard Gaussians; also $G(0)=0$. This finite-dimensional limit has no stochastically continuous version.

In particular it is not a fractional Brownian motion with a formal Hurst index zero. The analogous discrete-time spectral theorem of Deligiannidis and Utev (2013) assumes a strictly positive regular-variation index for the partial-sum variance; this zero-index endpoint lies outside its stated equivalence.

In fact allowing jumps does not rescue functional convergence: the normalized paths are not tight in Skorokhod $D[0,1]$ under either the $J_1$ or $M_1$ topology. Otherwise some subsequence would converge to a càdlàg process $Y$. The deterministic times at which a càdlàg process has a jump with positive probability form an at most countable set, so one can choose $t>0$ and a decreasing sequence $h_n\downarrow0$ such that $t$ and every $t+h_n$ are continuity times of $Y$.

Evaluation at such times is continuous in both topologies. The already identified finite-dimensional limit would therefore force

But right continuity of $Y$ forces the same increments to converge almost surely, hence in probability, to zero. This contradiction proves the claim. In particular the paths are not tight in $C[0,1]$.

The continuous-path obstruction is also visible directly. For every fixed $\delta>0$, stationarity gives

Since the increment is Gaussian, for every $\eta>0$ the probability that the uniform modulus over intervals of length $\delta$ exceeds $\eta$ has limit inferior at least $2\Phi(-\eta)$. Letting $\delta\downarrow0$ violates the necessary modulus-of-continuity condition for tightness. The endpoint therefore has a valid and explicit finite-dimensional limit but no càdlàg functional limit under this normalization.

A power-scale modulus profile

The failure of tightness has a sharper multiscale form. Suppose more specifically that the critical spectral law has

where $\ell(R^\theta)/\ell(R)\to1$ for each fixed $\theta>0$. Then the endpoint theorem and ordinary integration on the logarithmic scale give

For the shrinking lag $h_R=R^{-\beta}$, stationarity of the integrated process gives the exact reduction

whenever both times lie in the path interval. Consequently, for every fixed $\beta\ge0$,

Thus the correlation-time scale $R^{-1}$ is the continuity threshold, but the whole approach to it remembers the slowly varying critical law. Fixed positive lags correspond to $\beta=0$ and retain unit variance; lags at or below $R^{-1}$ vanish after normalization.

Between them lies a continuum of nonzero increment variances. The exponent $p+1$ also shows why there is no universal microscopic modulus profile under the bare critical hypothesis: different logarithmic spectral corrections have the same fixed-time Gaussian-dust limit but different power-scale paths to continuity.

An exact benchmark takes one-sided spectral density $f(\omega)=\omega e^{-\omega}$. Then $F(x)\sim x^2/2$, $C(t)=\operatorname{Re}(1-it)^{-2}$, and

Here $p=0$, so for $h_R=R^{-\beta}$ the finite-$R$ profile is exactly

The certificate checks this identity against covariance quadrature within $2.67\times10^{-15}$. On the time grid $0,0.1,0.25,0.5,0.75,1$, the maximum covariance-matrix error relative to the displayed limit falls from $0.332186$ at $R=2^{10}$ to $0.010381$ at $R=2^{320}$. At $\beta=(0.25,0.5,0.75,1,1.25)$, the maximum discrepancy from the limiting power-lag profile is below $1.6\times10^{-3}$ at $R=2^{320}$.

The deliberately slow logarithmic convergence is real: even for the short interval $\delta=0.01$, its normalized increment variance rises to $0.958476$ at $R=2^{160}$ and $0.979238$ at $R=2^{320}$, toward the nonzero limit one. At the terminal scale, the fixed dyadic lags $2^{-1},\ldots,2^{-10}$ retain normalized increment variances between $0.968750$ and $0.996875$; this is the finite-scale signature of the càdlàg obstruction.

Why Gaussianity is essential: an ergodic counterexample

The warning above is not merely about a hidden invariant random offset. Fix $0<\beta<1/2$ and let $G$ be a centered stationary unit-variance Gaussian process with covariance

This is a valid covariance because

a positive mixture of Ornstein–Uhlenbeck covariances. Since $r(t)\to0$, the Gaussian shift is mixing, hence ergodic. Its measurable transform

is therefore also stationary and ergodic. Wick's formula gives

Thus $X$ has the same covariance regular-variation exponent $\alpha=2\beta$ and the same candidate Hurst index $H=1-\beta$ as the Gaussian theorem. If $I_R=\int_0^R X_t\,dt$, then

Nevertheless the normalized limit cannot be Gaussian. A second Wick calculation gives

and symmetry of the ordered simplex yields

After $x=Rwq$, $y=Rw(1-q)$, dominated convergence gives

Consequently the standardized third cumulant converges to the strictly positive constant

Each normalized $I_R$ lies in the second Wiener chaos. Hypercontractivity therefore bounds its fourth moment uniformly, so convergence to a standard Gaussian would force its third moment to converge to zero, contradicting the display. The covariance exponent and even ergodicity do not identify the functional limit; the Hermite rank matters.

In the corresponding long-memory rank-two setting, Taqqu (1975) identifies the non-Gaussian Rosenblatt limit. For $\beta=0.3$, the deterministic certificate evaluates the finite-horizon variance and triple integral: its normalized third cumulant approaches $\gamma_{0.3}=2.067068891$ rather than zero.

An exactly covariance-matched Hermite hierarchy

The rank-two example is part of a sharper non-identification result. Fix one $\alpha\in(0,1)$. For each integer $q\geq1$, let $G^{(q)}$ be a centered stationary unit-variance Gaussian process with covariance

and let $H_q$ be the probabilists' Hermite polynomial of degree $q$. The normalized observable

is stationary and mixing, hence ergodic. The Gaussian–Hermite identity $\mathbb E[H_q(U)H_q(V)]=q!\rho^q$ for standard jointly Gaussian $(U,V)$ with correlation $\rho$ gives

Thus the whole hierarchy has exactly the same covariance at every lag, not merely the same asymptotic exponent. In particular, with $I_R^{(q)}=\int_0^R X_t^{(q)}dt$, every rank has the same variance and

Nevertheless, after normalization by this common variance, the integrated paths converge in $C[0,1]$ to different limits: rank one gives fractional Brownian motion, while rank $q\geq2$ gives the non-Gaussian rank-$q$ Hermite process. This is exactly the arbitrary-rank non-central limit theorem of Taqqu (1979), applied with $H=1-\alpha/2$. Hence no statistic determined only by the covariance or spectral measure can identify the functional limit, even within stationary mixing transforms of Gaussian processes.

For $\alpha=0.6$, the certificate checks ranks one through five by tensor Gauss–Hermite quadrature. Their covariance values agree within $4.44\times10^{-16}$; independent time quadrature checks the common integrated variance within $1.82\times10^{-12}$, and at $R=2^{24}$ its leading-term ratio is $0.998196069$.

To prove it, scale the Stieltjes measures by $\mu_\lambda(B)=\mu(\lambda B)/F(\lambda)$, where $\lambda=\varepsilon/T_\varepsilon=R_\varepsilon^{-1}$. Regular variation gives vague convergence to $\alpha x^{\alpha-1}dx$.

The kernel $2(1-\cos x)/x^2$ is bounded at zero; Potter bounds and $\alpha<2$ control its tail. Equivalently, the exact scale-collapse identity

shows that the normalized variance depends on $(T_\varepsilon,\varepsilon)$ only through $R_\varepsilon$. Frequencies bounded away from zero are negligible relative to the regularly varying term, and the limit uses

If the measure has a density $f(\omega)\sim a\omega^{\alpha-1}L(1/\omega)$, then $b=a/\alpha$ and the coefficient reduces to $a\pi/[\Gamma(3-\alpha)\sin(\pi\alpha/2)]$, the density formula. The cumulative-mass theorem also covers singular and pure-point spectra.

This single formula includes long memory for $0<\alpha<1$, ordinary Green–Kubo scaling at $\alpha=1$ when $L$ tends to a positive constant, and zero-Green–Kubo antipersistence for $1<\alpha<2$. The cumulative formulation also survives at the critical endpoint. If

then, still without assuming a density and for every joint path $T_\varepsilon/\varepsilon\to\infty$,

For the proof, put $R=T_\varepsilon/\varepsilon$ and $I(R)=\int_{(0,\infty)}(1-\cos(R\omega))\omega^{-2}\mu(d\omega)$. On $(0,R^{-1}]$, the inequality $1-\cos x\leq x^2/2$ gives an $O(L(R))$ contribution, while every interval bounded away from zero contributes $O(1)$. On $[R^{-1},\eta]$, Stieltjes integration by parts, followed by Dirichlet summation over the cosine half-periods, gives

Potter bounds make the remainder uniform as $R\to\infty$ and then $\eta\downarrow0$. Since the divergent borderline integral satisfies $L(R)=o(H(R))$, the last display is $2bH(R)\{1+o(1)\}$; substitution in the exact Fejér identity proves the factor $4b$. If a density exists with $f(\omega)\sim a\omega L(1/\omega)$, then $b=a/2$ and this reduces to the earlier $2a\varepsilon^2H(T/\varepsilon)$ law.

The divergence of $H$ is essential: when it stays bounded, the local equivalent of $F$ no longer determines a limiting coefficient. A finite global inverse spectral moment gives an order-$\varepsilon^2$ bound, but the coefficient converges only under the Fourier-decay condition stated below.

A concrete failure under mere stationarity is $X_t=Z+Y_t$, where $Z$ is a centered random constant with variance $\sigma_0^2$ and $Y$ is an independent stationary Ornstein–Uhlenbeck process with covariance $e^{-|t|}$. Then

The regular spectral density of $Y$ still produces the usual Green–Kubo term, but it is lower order than the invariant random offset. Dividing by $T^2$ gives the exact crossover $\sigma_0^2+2\{R-1+e^{-R}\}/R^2$. It tends to $\sigma_0^2+1=C(0)$ as $R\downarrow0$, equals $\sigma_0^2+2/e$ at $R=1$, and tends to $\sigma_0^2$ as $R\to\infty$.

The frequency-domain statement is stronger than an eventual-sign covariance theorem. For example, add a flat spectral band $0.8\mathbf1_{[4,5]}(\omega)$ to the Gamma density $e^{-\omega}\omega^{0.7}/\Gamma(1.7)$. Its covariance is

The second term makes the covariance change sign forever and dominate its pointwise tail, but it is separated from frequency zero and contributes only $O(\varepsilon^2)$. The theorem still gives the sharp $\varepsilon^{1.7}$ variance. This is the continuous-time counterpart of the spectral regular-variation criterion for partial sums proved by Deligiannidis and Utev (2013).

A density-free benchmark puts mass $n^{-\alpha-1}$ at frequency $1/n$. Its cumulative mass satisfies

so the theorem applies although the measure is purely atomic. For $\alpha=1.3$, the certificate evaluates the exact Fejér sum using a direct finite sum and an independent Hurwitz-zeta tail.

It finds order $1.300596$ and leading-constant ratio $1.000240$. The example also emphasizes that zero spectral mass at frequency zero is an observable-level averaging condition: atoms at nonzero frequencies need not disappear.

The same pure-point construction at the endpoint puts mass $n^{-3}$ at $1/n$. Then $F(x)\sim x^2/2$, so the critical theorem predicts $2\varepsilon^2\log(1/\varepsilon)$ at $T=1$. The direct Fejér sum and an independently refined Hurwitz-zeta tail agree within $2.71\times10^{-20}$.

Regressing $\operatorname{Var}A_\varepsilon/\varepsilon^2$ against $\log(1/\varepsilon)$ gives slope $1.966971$ against the limiting value $2$; at the smallest tested $\varepsilon=2^{-20}$ the leading-term ratio is $1.025526$. The deliberately slow logarithmic convergence distinguishes this endpoint from the power-law cases.

A separate joint-scale certificate evaluates the Gamma spectral benchmark along three paths with the same $R=T_\varepsilon/\varepsilon$: fixed maturity, $T_\varepsilon=R^{-1/2}\downarrow0$, and $T_\varepsilon=R^{1/2}\uparrow\infty$. After division by $T_\varepsilon^2$, all three exact variances agree to machine precision. At $R=2^{24}$, the power-law leading ratio is $0.992367$ and the critical logarithmic ratio is $1.000000000000$ on every path.

An independent covariance-quadrature certificate for the atom-plus-OU example checks the full $R$ crossover along maturities $0.2$, $1.3$, and $7$: normalized variances agree with the exact formula within $8\times10^{-15}$ and collapse across maturity paths within $5\times10^{-16}$, while the frozen, $R=1$, and invariant limits are respectively $1.369999980132$, $1.105758882343$, and $0.370000119209$. For the vector periodic example, independent covariance quadrature agrees with the matrix spectral formula within $2\times10^{-16}$ and verifies the rank pattern $2,1,2$.

Four covariance regimes

Integrable correlation

If $C$ is absolutely integrable and $K=\int_0^\infty C(s)ds$, dominated convergence gives, for fixed $T>0$,

This is the Green–Kubo regime used by the finite-chain pages. Exponential mixing is more than enough; a central limit theorem or a Poisson equation supplies the corresponding fluctuation limit.

Zero Green–Kubo and antipersistence

Integrability alone does not make the order-$\varepsilon$ coefficient nonzero. Suppose that $C$ is absolutely integrable,

and, for some $1<\alpha<2$,

with $L$ slowly varying and eventually positive. Karamata's tail theorem and its truncated-moment form give

Substitution in the exact identity yields the cancellation theorem

Since the left side is nonnegative, a nonzero regularly varying tail under these assumptions must have $c<0$. Negative long-lag covariance cancels the short-lag contribution exactly enough to replace the usual $\varepsilon$ variance by $\varepsilon^\alpha L(1/\varepsilon)$. The fluctuation amplitude is therefore $\varepsilon^{\alpha/2}L(1/\varepsilon)^{1/2}$: smaller than $\sqrt\varepsilon$, but generally larger than $\varepsilon$.

A fully valid stationary Gaussian benchmark has spectral density proportional to $e^{-|\omega|}|\omega|^{\alpha-1}$. Its normalized covariance is

It has $K=0$ exactly and $C_\alpha(t)\sim\cos(\pi\alpha/2)t^{-\alpha}$, whose coefficient is negative for $1<\alpha<2$. Direct integration gives the exact finite-$\varepsilon$ formula

This is the continuous-time analogue of the antipersistent fractional noises introduced by Mandelbrot and Van Ness (1968). It also shows why the statement “integrable covariance implies an order-$\varepsilon$ correction” requires the qualification $K\ne0$.

The second critical index and the $\varepsilon^2$ boundary

The cancellation theorem has a second critical index at $\alpha=2$. Keep $K=0$ and suppose

with $L$ slowly varying and eventually positive. Define

If $H(x)\to\infty$, Karamata's tail theorem gives $\int_x^\infty C(s)ds\sim cL(x)/x$, while its critical truncated-moment form and $L(x)=o(H(x))$ give $\int_0^x sC(s)ds\sim cH(x)$. Hence

Again nonnegativity forces $c<0$. Thus a pure $t^{-2}$ tail produces $-2c\varepsilon^2\log(T/\varepsilon)$, not a pure $\varepsilon^2$ law.

If instead the first covariance moment is absolutely integrable,

then $x\int_x^\infty|C(s)|ds\to0$ and the exact variance identity gives

For a nonzero stationary process the inequality is strict: $M<0$. To see this, use Bochner's spectral representation (1933)

Absolute integrability of $C$ makes $\mu$ absolutely continuous and therefore excludes a zero-frequency atom. For $B_R=\int_0^R X_tdt$, the covariance and spectral formulas give

Taking a Cesàro average in $R$, Fatou's lemma first proves integrability of $\omega^{-2}$ against $\mu$; dominated convergence then gives the exact identity

The right side is positive unless $\mu=0$, equivalently unless $X$ is zero in $L^2$. Thus the order-$\varepsilon^2$ coefficient cannot vanish under the stated assumptions. This includes regularly varying tails with $\alpha>2$, and the $\alpha=2$ case when $H(\infty)<\infty$.

There is a sharper purely spectral statement, and it identifies exactly what the absolute covariance-moment assumption supplies. Suppose only that the inverse spectral moment is finite,

For every joint path $R_\varepsilon=T_\varepsilon/\varepsilon\to\infty$, the Fejér identity is exactly

Consequently $0\leq\operatorname{Var}A_\varepsilon/(2\varepsilon^2)\leq2M_{-2}$, and convergence to $M_{-2}$ along every long-window path is equivalent to $\operatorname{Re}\widehat\nu(R)\to0$. It is sufficient that $\nu$ be Rajchman, meaning that its Fourier–Stieltjes transform vanishes at infinity.

In particular this holds whenever $\nu$ has an $L^1$ density, by the Riemann–Lebesgue lemma. The stronger time-domain assumptions above therefore imply a coefficient, but finiteness of $M_{-2}$ alone does not.

The obstruction is already a stationary Gaussian process with no zero-frequency atom. Let $X_t=U\cos t+V\sin t$ with independent standard Gaussian $U,V$. Its one-sided spectral measure is $\delta_1$, so $M_{-2}=1$, but

The normalized coefficient equals zero on $R_\varepsilon=2\pi n$ and two on $R_\varepsilon=(2n+1)\pi$. Thus a nonzero-frequency atom can survive at leading order at the saturated boundary even though it is negligible in the divergent $0<\alpha\leq2$ regimes. For comparison, if $\nu(d\omega)=e^{-\omega}d\omega$ (equivalently $\mu(d\omega)=\omega^2e^{-\omega}d\omega$), then

The numerical certificate evaluates this continuous identity independently by quadrature and checks both exact subsequences of the pure-tone counterexample.

Even when the pointwise coefficient fails to converge, averaging over the number of observed fast time units has an exact limit. Put

Then Wiener's continuous Fourier–Stieltjes lemma gives

Here the sum is over the atoms of the finite weighted measure $\nu$. The proof is direct.

The average of $\cos(R\omega)$ tends to zero for each $\omega>0$. For the square, expand the product and use

Dominated convergence against $\nu\otimes\nu$ leaves half the mass of the diagonal, namely half the sum of squared atom masses. Thus the normalized coefficient converges to $M_{-2}$ in Cesàro mean square if and only if $\nu$ is nonatomic. This is strictly weaker than pointwise Rajchman decay.

For example, a shifted middle-thirds Cantor measure is nonatomic. If $Y=1+\sum_{k\geq1}2\xi_k3^{-k}$ for independent fair bits $\xi_k$, then

At $R=2\pi3^n$ this has the same nonzero absolute value for every $n$. Its pointwise coefficient therefore fails to settle, while its Cesàro mean-square error still vanishes.

For the pure tone, the ordinary coefficient $1-\cos R$ oscillates forever, but its Cesàro mean is one and its mean-square deviation from one is $1/2$. The certificate also uses atoms of masses $0.4$ and $0.25$ at frequencies $1$ and $\sqrt2$: the predicted mean-square limit is $(0.4^2+0.25^2)/2=0.11125$, independently of their incommensurate locations.

The same Gamma-spectral benchmark makes both boundaries exact. At $\alpha=2$ and $\alpha>2$, respectively,

For the second formula, $M=-1/\{(\alpha-1)(\alpha-2)\}$. Together with the $1<\alpha<2$ theorem, this is the complete zero-Green–Kubo trichotomy under the stated regular-variation or first-moment assumptions; it is distinct from the oscillatory periodic example below.

Long-range dependence

More generally, suppose that the covariance is eventually positive and regularly varying:

where $L$ is slowly varying. Karamata's integral theorem gives

Substitution in the exact covariance identity, with $x=T/\varepsilon$, therefore gives

The Green–Kubo integral diverges and the fluctuation size is $\varepsilon^{\alpha/2}L(1/\varepsilon)^{1/2}$ rather than $\sqrt\varepsilon$. The pure-power formula is the special case $L\equiv1$.

The critical index is not one formula

At the boundary $C(t)\sim cL(t)/t$, put

If $H(x)\to\infty$, then Karamata's theorem and $L(x)=o(H(x))$ give

If $H(\infty)<\infty$, the tail is integrable and the ordinary Green–Kubo order $\varepsilon$ returns. Thus $C(t)\sim c/t$ gives $2cT\varepsilon\log(1/\varepsilon)$, but a logarithmic modifier changes even the critical scale. For $L(t)=(\log t)^\beta$,

whereas $\beta<-1$ is integrable. Therefore neither the coefficient nor even the power-and-log scale of the correction is universal.

The certificate tests both new scales on the valid covariance

It is a positive mixture of Ornstein–Uhlenbeck covariances. Numerical integration of the exact finite-horizon variance gives asymptotic ratios $1.02166$ for $\alpha=0.4$ and $0.94394$ at the critical $\varepsilon\log^2(1/\varepsilon)$ scale.

For the invariant-atom counterexample, independent covariance quadrature agrees with the atom-plus-OU formula within $3.33\times10^{-16}$. At the smallest tested $\varepsilon$, the normalized variance is $0.370093896$ against the limiting atom mass $0.37$, while the residual after subtracting the atom is $0.999953050$ of its Green–Kubo leading term.

For the zero-Green–Kubo benchmark, independent covariance quadrature agrees with the exact formula within $1.1\times10^{-17}$. At $\alpha=1.4$, the measured order and asymptotic ratio are $1.399942$ and $0.999921$; at $\alpha=1.7$, they are $1.697433$ and $0.992367$. The boundary checks obtain the exact log-corrected ratio $1.000000000000$ at $\alpha=2$.

At $\alpha=3$, independent quadrature gives $M=-0.5$, measured order $2.000000000$, and asymptotic ratio $0.999999999996$. Independent spectral quadrature gives $\int\omega^{-2}\mu(d\omega)=0.5$, agreeing with $-M$ within $1.1\times10^{-16}$.

The low-frequency certificate also reconciles the spectral and covariance-tail constants within $5.33\times10^{-15}$. For the oscillatory-band example above, independent covariance and spectral integration agree within $2.95\times10^{-17}$; the measured order is $1.697454$, the asymptotic ratio is $0.992431$, and the covariance changes sign 3,151 times on the deterministic mesh $20\le t\le2000$.

For the pure-point spectrum, the cumulative-mass ratio at $x=2^{-20}$ is $1.000000620$. Two independently truncated Fejér sums agree within $8.67\times10^{-19}$; the measured variance order and asymptotic ratio are $1.300596$ and $1.000240$.

For the pure-point critical spectrum, the cumulative-mass ratio is $1.000000954$, the independent summation discrepancy is $2.71\times10^{-20}$, the fitted logarithmic slope is $1.966971$, and the smallest-$\varepsilon$ leading-term ratio is $1.025526$.

A bounded positive volatility counterexample

For $0<\alpha<1$, let $G_t$ be a centered stationary Gaussian process with correlation

This is a valid covariance because it is a positive Gamma mixture of Gaussian kernels. Its spectral measure has no atoms, so the process is ergodic. Define a bounded strictly positive variance by

The Gaussian sign identity gives

Consequently, for integrated variance $V_\varepsilon(T)=\int_0^T v_{t/\varepsilon}dt$,

The observable is bounded and the driver is stationary and ergodic, yet the order-$\varepsilon$ Green–Kubo smile does not apply.

Implication for the uncorrelated volatility smile

When the Brownian return is independent of the variance driver, conditioning on integrated variance gives the Hull–White mixture $\mathbb E[C_{\rm BS}(V_\varepsilon)]$. For a smooth payoff, a second-order delta method gives

provided the normalized integrated variance has the required moment convergence. The sign-Gaussian example has Hermite rank one, and the long-range Gaussian limit theory supplies that normalization. Its convexity smile is therefore of order $\varepsilon^\alpha$, not $\varepsilon$.

This statement concerns the uncorrelated variance-mixture effect. Correlation between the stock and a fast diffusive factor can separately create an order-$\sqrt\varepsilon$ skew term; the two mechanisms should not be conflated.

Ergodic need not mean diffusive

Let $U$ be uniform on the circle and $Y_t=U+t/\varepsilon\pmod{2\pi}$. The continuous flow is stationary and ergodic. With $v(y)=v_0+\delta\cos y>0$,

and hence exactly

The correction is order $\varepsilon^2$ with an oscillatory coefficient. The one-sided covariance integral does not converge. This example is smooth, bounded and positive, so stronger mixing or fluctuation assumptions really are necessary.

Scope and literature

The theorems above determine the variance scale from either the covariance tail or the low-frequency spectral density, together with whether the Green–Kubo integral vanishes; they do not claim a universal option expansion for every long-memory process or nonsmooth payoff. Distributional limits depend on the process and on the Hermite rank of the observable. The regular-variation step is Karamata's integral theorem (1933).

Primary long-memory references are Taqqu (1975), Dobrushin and Major (1979), and Taqqu (1979). In finance, Comte and Renault (1998) develop continuous-time long-memory stochastic volatility.

Sircar and Papanicolaou's fast-volatility calculation invokes a fluctuation limit in addition to ergodicity. The finite-state chain used on this site satisfies the stronger assumptions; the counterexamples delimit the generality of the claim rather than contradicting the chain calculation.

Certificate: verify_long_memory.py.